Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7021363 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,6,10,14)(4,7,12,16)(9,13,11,15)(17,18,23,29,24,30,22,28)(19,31,33,42,35,45,34,25)(20,36,37,26,38,43,21,27)(32,44,39,48,47,40,46,41);(1,3)(2,6)(4,11)(5,10)(7,15)(8,14)(9,12)(13,16)(17,19)(18,25)(20,32)(21,39)(22,33)(23,34)(24,35)(26,40)(27,44)(28,31)(29,45)(30,42)(36,41)(37,46)(38,47)(43,48);(1,4)(2,7)(3,9)(5,12)(6,13)(8,16)(10,11)(14,15)(17,20,23,37,24,38,22,21)(18,26,28,36,30,27,29,43)(19,32,34,46,35,47,33,39)(25,40,31,41,42,44,45,48);(4,12)(7,16)(9,11)(13,15)(17,21,22,38,24,37,23,20)(18,27,28,43,30,26,29,36)(19,32,34,46,35,47,33,39)(25,41,45,40,42,48,31,44);(2,8)(3,10)(7,16)(9,11)(17,22,24,23)(18,28,30,29)(19,33,35,34)(20,37,38,21)(25,31,42,45)(26,43,27,36)(32,46,47,39)(40,48,44,41)Polycyclic presentation:
2^9|a^2=fh,c^2=g,d^2=fi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6949285);